LET F q be the nite eld with q elements and F n q be the

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1 JOURNAL OF L A TEX CLASS FILES, VOL., NO., NOVEMBER 2002 A Classi cation of Posets admitting MacWilliams Identity Hyun Kwang Kim and Dong Yeol Oh Abstract In this paper all poset structures are classi ed which admit the MacWilliams identity, and the MacWilliams identities for poset weight enumerators corresponding to such posets are derived. We prove that being a hierarchical poset is a necessary and suf cient condition for a poset to admit MacWilliams identity. An explicit relation is also derived between P-weight distribution of a hierarchical poset code and P-weight distribution of the dual code. Index Terms MacWilliams identity, poset codes, P-weight enumerator, leveled P-weight enumerator, hierarchical poset. I. INTRODUCTION LET F be the nite eld with elements and F n be the vector space of n-tuples over F. Coding theory may be considered as the study of F n when F n is endowed with Hamming metric. Since the late 980 s several attempts have been made to generalize the classical problems of the coding theory by introducing a new non-hamming metric on F n cf [8-0]). These attempts led Brualdi et al. [] to introduce the concept of poset codes. First, we begin by brie y introducing the basic notions of poset code such as poset-weight and posetdistance. See [] for details. Let F n be the vector space of n-tuples over a nite eld F with elements. Let P be a partial ordered set, which will be abbreviated as a poset in the seuel, on the underlying set [n] =, 2,...,n} of coordinate positions of vectors in F n with the partial order relation denoted by as usual. For u = u,u 2,,u n ) F n, the support suppu) and P-weight w P u) of u are de ned to be suppu) =i ui 0} and w P u) = <suppu) >, where <suppu) > is the smallest ideal recall that a subset I of P is an ideal if a I and b a, then b I) containing the support of u. It is well-known that for any u, v F n, d P u, v) :=w P u v) is a metric on F n. The metric d P is called P-metric on F n. Let F n be endowed with P-metric. Then a linear) code C F n is called a linear) P-code of length n. The P-weight enumerator of a linear P-code C is de ned by W C,P x) = x wpu) = n A i,p x i, where A i,p = u C wp u) =i}. This research was supported by the Com 2 MaC-KOSEF and POSTECH BSRI research fund. H.K.Kim and D.Y.Oh are with the Department of Mathematics, Pohang University of Science and Technology, Pohang , Korea. Remark : If P is an antichain, then P-metric is eual to Hamming metric. So P-weight enumerator of a linear code C becomes Hamming weight enumerator of C. The MacWilliams identity for linear codes over F is one of the most important identities in the coding theory, and it expresses Hamming weight enumerator of the dual code C of a linear code C over F in terms of Hamming weight enumerator of C. Since Hamming metric is a special case of poset metrics, it is natural to attempt to obtain MacWilliamstype identity for certain P-weight enumerators. See [3-5] for this direction of researches. Essentially, what enables us to obtain MacWilliams identity for Hamming metric is that Hamming weight enumerator of the dual code C is uniuely determined by that of C. The following example suggests that we need some modi cation to generalize MacWilliams identity for certain type of poset weight enumerators. Example.: Let P =, 2, 3} be a poset with order relation < 2 < 3 and P =, 2, 3} be a poset with order relation > 2 > 3. Consider the following binary linear P- codes: C = 0, 0, 0), 0, 0, )}, C 2 = 0, 0, 0),,, )}. It is easy to check that P-weight enumerators of C and C 2 are given by W C,Px) =+x 3 = W C2,Px). The dual codes of C and C 2 are respectively given by C = 0, 0, 0),, 0, 0), 0,, 0),,, 0)} and C2 = 0, 0, 0),,, 0),, 0, ), 0,, )}. The P-weight enumerators of C and C2 are given by W C,Px) =+x +2x 2, W C 2,Px) =+x 2 +2x 3, while P-weight enumerators of C and C2 are given by W C,P x) =+x2 +2x 3 = W C 2,P x). As it is seen above, although P-weight enumerators of the codes C and C 2 are the same, P-weight of the dual codes may be different. Fortunately,however, P-weight enumerators of the dual codes are the same. Feeding back this information we de ne, for a given poset P, the poset P as follows: P and P have the same underlying set and x y in P y x in P. The poset P is called the dual poset of P. De nition.2: Let P be a poset on [n]. It is said that P admits MacWilliams identity if P-weight enumerator of the dual code C of a linear code C over F is uniuely determined by P-weight enumerator of C. For an illustration of our de nition, we give two classes of posets which admit MacWilliams identity.

2 JOURNAL OF L A TEX CLASS FILES, VOL., NO., NOVEMBER In [], Rosenbloom and Tsfasman introduced a new non-hamming metric which is called the ρ-metric or the Rosenbloom-Tsfasman metric on linear spaces over - nite elds. The ρ-metric is de ned on the linear space Mat m,n F ), where Mat m,n F ) is the set of all matrices with m-rows and n-columns over F. For the sake of simplicity, we introduce it only in the case m =and refer to [2], [2] for a general treatment. We remark that ρ-metric can be realized as a poset metric over the disjoint union of chains. Now let m =.Foru =u,u 2,,u n ) F n,weset ρ0) = 0 and ρu) =maxi ui 0} for u 0.Foragiven linear code C F n, we de ne the ρ-weight enumerator for C by W C z) = n w i C)z i = z ρu), where w i C) = u C ρu) =i}, 0 i n. The following identity was obtained in [2, Theorem 4.4]: z )W C z)+ z = C z n+ [ z)w C )+z ], ) z where C = v F n <u,v>=0for all u C}, and <u,v>= n u i v n+ i. i= If we put P =, 2,...,n} with order relation < 2 <... < n, then ρ-metric becomes P-metric and W C z) = W C,P z). The MacWilliams identity for Hamming weight enumerators and the work of Skriganov [2, Theorem 4.4] state that antichain and chain on [n],n, admit MacWilliams identity. In this paper, we classify all poset structures which admit MacWilliams identity. We also derive MacWilliams identities for poset weight enumerators corresponding to such poset codes. Section 2 gives a necessary condition for a poset P to admit MacWilliams identity. It will be proved that being a hierarchical poset is a necessary condition for a poset P to admit MacWilliams identity. In section 3, MacWilliams identity for a hierarchical poset code is derived, and it will be proved that our necessary condition in Section 2 is also a suf cient condition for admitting MacWilliams identity. Section 4 examines the relationship between A i,p },...,n and A } i,p,...,n. More precisely, we will express explicitly A in terms of A i,p j,p, 0 j n, using Krawtchouk polynomials. II. NECESSARY CONDITION FOR ADMITTING MACWILLIAMS IDENTITY In this section, we will give a necessary condition for a poset P to admit MacWilliams identity. First, a hierarchical poset as the ordinal sum of antichains is introduced, and it will be proved that being a hierarchical poset is a necessary condition for a poset P to admit MacWilliams identity. Let n,n 2,...,n t be positive integers with n + n n t = n. We de ne the poset Hn; n,n 2,...,n t ) on the set i, j) i t, j ni } whose order relation is given by i, j) < l, m) i<l. The poset Hn; n,n 2,...,n t ) is called a hierarchical poset with t-levels and n-elements. For each i t, the subset i, j) j ni } of Hn; n,n 2,...,n t ) is called i th - level set of Hn; n,n 2,...,n t ), and it is denoted by Γ i H). Note that Γ i H) is an antichain with cardinality n i. Let Hn; n,n 2,...,n t ) be a hierarchical poset with t- levels and n-elements. From now on, we will identify the underlying set of Hn; n,n 2,...,n t ) with the coordinate positions of vectors in F n by identifying the subset n + n n i +,...,n + n n i + n i } of [n] with the i th level set Γ i H) in an obvious way. By convention we set n 0 =0. For a poset P, we de ne minp) = i P i is minimal in P}. The following lemma is an immediate conseuence of the concepts developed so far and will be useful in the seuel. Lemma 2.: Let P be a poset on [n] and P be the dual poset of P. Foru F n,wehave w P u) =n suppu) minp). For a given poset P, we put P = P \ minp). Then P is also a poset under the partial order relation induced from that of P. Lemma 2.2: Let P be a poset of cardinality n. Suppose that minp) has n elements. Then, for each vector u F n satisfying suppu) minp), n n v F n u v =0 and wp v) =n}, where a b denotes that a divides b. Proof : Without loss of generality, we may assume that minp) =, 2,...,n }. Since suppu) minp), u can be written in the form u =a,...,a i, 0,...,0), where 0 a j F for all j i and i n. Let A be the set of vectors over F of length i de ned by A := b,...,b i ) F i a b + + a i b i =0and b j 0 for j i}. Then we have v F n u v =0,w P v) =n} = A n n ) n i. Lemma 2.3: Suppose that P admits MacWilliams identity. Then, for each minimal element i in P = P \ minp) and j in minp), wehavei j. Proof : Let P = n and minp) = n.ifn = n, then the lemma is true. Hence we may assume that n>n. We claim that <i> =+ minp) for each i minp ). Suppose not. Then we can choose i minp ) such that <i> < + minp). Since <i> < + minp), we can choose two vectors u,u 2 F n such that suppu )= i},suppu 2 ) minp), and < suppu ) > = < suppu 2 ) >. Now we consider two linear codes C and C 2 generated by u and u 2, respectively. Since <suppu ) > = <suppu 2 ) >, C and C 2 have the same P-weight enumerator. It follows from our assumption that C and C2 have the same P-weight enumerator. Therefore we should have the following euation: v C wp v) =n} = v C2 wp v) =n}.

3 JOURNAL OF L A TEX CLASS FILES, VOL., NO., NOVEMBER It is immediate that v C wp v) =n} = n n+) ) n, and it follows from Lemma 2.2 that n n v C 2 wp v) =n}. These yield that n n n n +) ) n. However it is impossible, since is power of a prime. This prove that < i> =+ minp) for each i minp ). The statement of Lemma 2.3 follows immediately from this fact. Remark : If i P, then i k for some k minp ). Therefore we have obtained : if P admits MacWilliams identity, then for i P and j minp), we have i j. Lemma 2.4: If a poset P admits MacWilliams identity, then P also admits MacWilliams identity. Proof : Let P = n and minp) = n.ifn = n, then the lemma is true. Hence we may assume that n>n. Let C, C 2 be two linear codes of length n n with the same P -weight enumerators. We consider two linear codes of length n de ned by C i = F n C i := u, v) u F n,v C i },i=, 2. It follows from the previous remark that C and C 2 have the same P-weight enumerators. Therefore C, C2 have the same P-weight enumerators. Since Ci = u, v) u = 0 F n,v C i }, for i =, 2, C and C 2 have the same P -weight enumerators. This proves that P also admits MacWilliams identity. From the above lemmas and inductive argument, we have the following theorem. Theorem 2.5: If P admits MacWilliams identity, then P is a hierarchical poset. III. MACWILLIAMS IDENTITY FOR A HIERARCHICAL POSET CODE In this section, we will derive the MacWilliams identity for a hierarchical poset code. Let C be a linear P-code of length n over F. We rst introduce the leveled P-weight enumerator W C,P x : y 0,y,...,y t ) and obtain an euation which relates W C,P x : z t+,z t,...,z ) with variations of leveled P- weight enumerator of C. By specializing this euation, we will obtain the MacWilliams identity for a hierarchical poset code, and prove that our necessary condition in Section 2 is also a suf cient condition for admitting the MacWilliams identity. In this section, P will denote a hierarchical poset with t-levels and n-elements unless otherwise speci ed. Let P = Hn; n,n 2,...,n t ) be a hierarchical poset with t-levels and n - elements on the set [n] =, 2,...,n}. As mentioned earlier, we identify the underlying set of P with the coordinate positions of vectors in F n. Since n = n + + n t and F n = F n F n 2 F n t, for u F n, we may write u =u,u 2,...,u t ), and u i F ni. For an integer 0 i t, we also use the following notation: n i = n n + + n i )=n + + n t, ũ =u,...,u t ) F ni. For a linear P-code C, we de ne C i and C i as follows: C i = u C u = = u t =0}, and Ci = u C i ui 0}. Let C be a linear P-code of length n over F. We introduce the leveled P-weight enumerator W C,P x : y 0,y,...,y t ) of C as follows: W C,P x : y 0,y,...,y t )= x wp u) y sp u) = A 0,P y 0 +A,P x + + A n,px n )y +A n+,px n+ + + A n+n 2,Px n+n2 )y 2 + +A n+ +n t +,Px n+ +nt + + +A n+ +n t,px n+ +nt )y t, where s P u) = maxi u i 0} in the expression u = u,...,u t ) and A i,p = u C wp u) =i}. For the sake of simplicity in our calculation, we also introduce the i th -level P-weight enumerator C,P x), i t, as follows: ni C,P x) := A n+ +n i +j,px n+ +ni +j =A n+ +n i +,Px + + A n+ +n i,px ni )x n ni. By convention, we put LW 0) C,P x) :=A 0,P. Remark : a) If we put y 0 = y = = y t =, then the leveled P-weight enumerator of C becomes the usual P-weight enumerator of C: W C,P x :,...,) = W C,P x) = t C,P x). 2) b) If we put y j =for j i and y k =0for k>i, then the leveled P-weight enumerator of C becomes the P-weight enumerator of the subspace C i c) It is easy to see that W Ci,Px) W Ci,Px) = C,P x) = x wpu). 3) i Recall that an additive character χ on F is just a homomorphism from the additive group of F into the multiplicative group of complex numbers of magnitude. We give the following lemmas about additive characters on F which play an important role in the proof of the main theorem without proof. See [6], [7] for detailed discussion on additive characters. Lemma 3.: Let χ be a nontrivial additive character of F and α be a xed element of F. Then if α =0 χαβ) = 0 if α 0. β F Lemma 3.2: Let χ be a nontrivial additive character of F. Then, for any linear code C over F, χu v) = v C 0 if u C if u C. Let f be a complex-valued function de ned on F n. The Hadamard transform f of f is de ned by

4 JOURNAL OF L A TEX CLASS FILES, VOL., NO., NOVEMBER fu) = v F n χu v)fv). The following lemma, which is called the discrete Poisson summation formula, is an easy conseuence of Lemma 3.2. Lemma 3.3: Let C be a linear code of length n over F and f be a function on F n. Then fu) = fu). Lemma 3.4: If a function f is de ned on F n by fu) = x whu), then its Hadamard transform f of f is given by fu) = v F n χu v)fv) = + )x) n wh u) x) wh u). The MacWilliams identity for Hamming weight enumerators can be obtained by applying discrete Poisson summation formula to the complex-valued function fu) = x whu). We now apply discrete Poisson summation formula to the complex-valued function fu) =x w P u) z sp u), where s P u) = mini u i 0} in the expression u = u,...,u t ),u i F ni. By convention, we set s P 0) = t +. We now analyze the value fu) in detail. For an integer 0 i t, we put B i = u =u,...,u t ) F n 0, and u 0}. t Note that F n = B i is a disjoint union. It follows from the above observation that fu) = χu v)fv) v F n u =...= u i = t = χu v)x wpv) z sp v). 4) v B i Denote the inner sum in 4) by S i u), 0 i t. Forv B i with i<t,wehavew P v) =n i n t + w H v )= n + w H v ) and s P v) =i +, where n i = n n + n 2 + +n i ).Forv F n, we write v =v,v 2,,v i, ṽ ), where ṽ =v,v i+2,...,v t ) F ni. Hence the inner sum S i u) in 4) for i<tis S i u) = χu v)x wpv) z sp v) v B i = x n z χu v)x whv) v B i = x n z χũ i+2 ṽ i+2 ) v 0 F n v i+2 F n χu v )x wh v). It follows from Lemma 3.4 that χu v )x whv) v 0 F n = x Qx) )n Qx) wh u), where Qx) = S i u) = x n z x + )x. Hence we have v i+2 F n χũ i+2 ṽ i+2 ) x Qx) )n Qx) wh u) ) = x n x z Qx) )n Qx) wh u) ) χũ i+2 ṽ i+2 ). v i+2 F n For i<t, it follows from the Lemma 3.2 that 0 if ũ i+2 0 F n S i u) = x) n z x Qx) )n Qx) wh u) ) if ũ i+2 =0. 5) For i = t, it is clear that S t u) =z t+. Hence we have fu) =z t+ + t S i u), where S i u) is given by 5). Let C be a linear P-code of length n over F, where P = Hn : n,...,n t ) is a hierarchical poset with t-levels and n- elements. For 0 i t, we consider the subspace C i of C de ned by C i = u =u,...,u t ) C u = = u t =0}. Note that C i is the subset of the codewords u of C satisfying ũ =0. Therefore it follows from 5) that S i u) = =x) n z S i u) x Qx) )n Qx) wh u) 6) ). Denote the right hand side of the sum in 6) by SC ). Then, SC ) = x Qx) )n Qx) wh u) ) =+ )x) n Qx) wh u) C.7) Put C 0 = u C u =0} and C = u C u 0}. For each u C,wehave w P u) =w H u )+n +n 2 + n i = w H u )+n n i ). It follows from this observation that the inner sum in 7) is + )x x ) n ni x + )x )w Pu) + C i.

5 JOURNAL OF L A TEX CLASS FILES, VOL., NO., NOVEMBER It follows form 2) and 3) that S i u) = S i u) Since fu) = we have =x) n + )x) n z Qx) w Pu) Qx) )n ni +x) n z Ci + )x) n C ) = x + )x x )n ) n n x) ni x LW ) C,P Qx))z +x) n z Ci + )x) n C ).8) t χu v)x w P v) z sp v) = z t+ + t S i u), v B i fu) = z t+ + t S i u) = z t+ + x t x )n t + t a i x)lw ) C,P Qx))z b i x) C i z x) n C z, 9) where a i x) = + )x x )x) n x) n. Since W C,P x : y 0,...,y t ) = x + )x, and a ix) = + )x x ) n ) n n x) ni and b i x) =+ t C,P x)y i, Qx) = n x) ni, the rst summation in 9) becomes x x x )n W C,P + )x : f 0,f,...,f t ), 0) where 0 if i =0 f i = + )x x ) n ni x) ) z i if i. Since C i = A 0,P +A,P + + A n,p) + + A n+ +n i +,P + + A n+ +n i,p), we have the following euation: t b i x) C i z = b 0 x) C 0 z + b x) C z b t x) C t z t = A 0,P b 0 x)z + b x)z b t x)z t ) +A,P + + A n,p)b x)z b t x)z t ) + + +A n+ +n t 2+,P + + A n+ +n t,p)b t x)z t. Let g j = t i=j b i x)z, for 0 j t and g t =0. Recall that b i x) =x) n + )x) n.) Since C,P ) = C i C i = A n+ +n i +,P+ + A n+ +n i,p and W C,P : y 0,...,y t ) = t C,P )y i, the second summation in 9) becomes t t b i x) C i z = C,P )g i = W C,P : g 0,g,...,g t ), 2) where t x) n + )x) n z if 0 j t g j = i=j 0 if j = t. 3) In the same manner, the last summation in 9) becomes t x) z C = W C,P : h 0,h,...,h t ),4) where t x) ni z i if j t i=j h j = t x) ni z i if j =0. i= 5) By applying discrete Poisson summation formula fu) = fu), we nally obtain the following theorem. Theorem 3.5: Let P = Hn : n,...,n t ) be the hierarchical poset of n-elements with t-levels and C be a linear P-code of length n over F. Then W C,P x : z t+,...,z )= fu) = z t+ + x x )n W C,P Qx) :f 0,...,f t ) +W C,P : g 0,...,g t ) W C,P : h 0,...,h t ) ), where Qx) = x + )x, and f i,g i,h i are given by Euations ), 3) and 5). If we put z = z 2 = = z t+ =in Theorem 3.5, then W C,Px :,,...,) becomes the usual the P-weight enumerator W C,Px) of the dual code C on the poset P. Hence the P-weight enumerator of the dual code C is uniuely determined by the P-weight enumerator of C itself. Combining this with Theorem 2.5, we obtain the following main theorem. Theorem 3.6: A poset P admits MacWilliams identity if and only if P is a hierarchical poset. As an illustration, we apply Theorem 3.5 to special cases, and compare our results with the previous result. Let P be an antichain of n-elements, that is, P is a hierarchical poset with -level. Put z = z 2 =. The euations ), 3) and 5) can be written as follows:

6 JOURNAL OF L A TEX CLASS FILES, VOL., NO., NOVEMBER f 0 =0, f = + )x ) n x) n, 6) x g 0 =+ )x) n, g =0, 7) h 0 = h =. 8) After a simple calculation, we obtain the following corollary. Corollary 3.7: Let P be an anti-chain of n-elements and C be a linear P-code over F. Then, W C x) =W C,Px :, ) = + x ) )x)n W C. 9) + )x We remark that 9) is exactly the classical MacWilliams identity for Hamming weight enumerators cf [7, Ch5, Theorem 3]). Let P be a chain of t-elements, that is, P is a hierarchical poset of t-levels and n = = n t =so that n i = t i for 0 i t. Put z = z 2 =...= z t =. Then we have the following euations: 0 if i =0 f i = x) t+ + )x ) i x x) if i t, 20) g i = + )x x) t i ) if 0 i t, 2) x x h i = x)t ) if i t x x)t ) if i t. 22) From 20), 2), and 22), we have the followings: x x )t W C,P x + )x : f 0,...,f t ) = x)x) t W C,P x ) ), 23) W C,P : g 0,...,g t ) + )x = x) t W C,P x x ) ), 24) W C,P : h 0,...,h t ) = x)t+ x W C,P x ) x x)t. 25) By applying 23) 25) to Theorem 3.5, we have the followings: W C,P x) =W C,Px :,,...,) )x = x + x) t+ x) W C,P ) x x )+xx)t. 26) Note that C = t and some computations yield the following corollary. Corollary 3.8: Let P be a chain of n-elements and C a linear P-code over F. Then, x )W C,P x)+ x = C x t+ x)w C,P x )+x ). 27) This is the same as the result in [2, Theorem 4.4]. IV. RELATIONSHIP BETWEEN WEIGHT DISTRIBUTIONS Let P = Hn; n,...,n t ) be a hierarchical poset of n- elements with t-levels and P be its dual poset. Let C be a linear P-code of length n over F, and let A i,p },...,n resp. A } i,p,...,n) be the weight distributions of the Presp. P) -code C resp. C ), that is, A i,p = u C wp u) =i} while A = v i,p C w P v) = i}. In this section, we will study the relationship between A i,p },...,n and A } i,p,...,n. More precisely, we will express explicitly A i,p in terms of A j,p, 0 j n, using Krawtchouk polynomials. Before proceeding with hierarchical posets, we brie y review the relationship between A i },...,n and A i },...,n, where A i = u C w H u) =i} and A i = u C w H u) = i}. For convenience, we set γ = in this section. De nition 4.: For any prime power and positive integer n, the Krawtchouk polynomial is de ned by k ) ) x n x P k x : n) = ) j γ k j,k =0,,...,n. j k j j=0 These polynomials have the generating function ) n i n +γx x) i = P k i : n)x k, 0 i n. 28) Theorem 4.2: Relationship between Hamming weight distributions) Let C be a linear code of length n over F. Then A k = n A i P k i : n), where A k = u C wh u) =k} and A i = u C w H u) =i}. Let P = Hn; n,...,n t ) be a hierarchical poset of n- elements with t-levels and C be a linear P-code of length n over F. We de ne C,P x, y) as follows: ni C,P x, y) := A n+ +n i +jx ni j y n+ +ni +j. 29) Then it is easy to see that C,P x, y) =W C i,px, y) x ni W Ci,Px, y). 30) The C,P x, y) is also called the ith level P-weight enumerator of C. By setting z = z 2 = = z t+ =in Theorem 3.5, we obtain the following theorem.

7 JOURNAL OF L A TEX CLASS FILES, VOL., NO., NOVEMBER Theorem 4.3: Let P = Hn; n,...,n t ) and C be a linear P-code over F. Then W C,P x) =+ t x) n x) ) LW n ni C,P + γx, x) + t x) n + γx) n C i C ). 3) Since n n i = n + + n i, the following euation can be easily derived from 28), 29), and 30): x) n x) ) LW n ni C,P + γx, x) n = x) n n A n+ +n i+jp k j : n ) ) x k. For convenience, we set a k j : n ):= n Since P 0 j : n )=,wehave a 0 j : n )= n A n+ +n i+jp k j : n ). A n+ +n i+j = C C i. 32) Therefore, the rst summation in 3) becomes t x) t x) n n a k j : n )x k ). 33) It follows from the binomial series that the last summation in 3) becomes n ) n n C i C + γ k C i x k ). 34) k= By 32), 33), and 34), the RHS of 3) in the Theorem 4.3 becomes + t x) n n k= ak j : n )+ n k k ) γ k C i ) x k. 35) On the other hand, the LHS of 3) in the Theorem 4.3 can be written as W C,P x) = A 0,P + A,P x + + A n t,p xnt + A n t+,p x + + A n t+n t,p xnt ) x nt + + A n x + + ) A t+ +n 2+,P n t+ +n,p x xn nt+ +n2 t =+ n x n k= A n t+ +n i+2+k,p xk. 36) n Since a k j : n )= A n+ +n i+j,pp k j : n ) and C i = n+ +ni A k, we have the following theorem from 35) and 36). Note A 0,P = A 0,P =.) Theorem 4.4: Let P = Hn; n,...,n t ) be a hierarchical poset of n-elements with t-levels and C be a linear P-code of length n over F. Then, for each 0 i t, k n, A n t+ n i+2+k,p = n n n n + k P k j : n )A n+ +n i+j,p n + +n i )γ k j=0 REFERENCES A j,p. [] R. A. Brualdi, J. Graves and K. M. Lawrence, Codes with a poset metric, Discrete Math ) [2] S. T. Dougherty and M. M. Skriganov, MacWilliams duality and the Rosenbloom-Tsfasman metric, Moscow Math. J ), no, [3] J. N. Gutiérrez and H. Tapia-Recillas, A MacWilliams identity for posetcodes, Congr. Numer ) [4] Y. Jang and J. Park, On a MacWilliams identity and a perfectness for a binary linear n, n,j)-poset codes, Discrete Math ) [5] D. S. Kim and J. G. Lee, A MacWilliams-type identity for linear codes on weak order, Discrete Math ) [6] R. Lidl and H. Niederreiter, Introduction to nite elds and their applications, Cambridge University Press, Cambridge 994. [7] F. J. MacWilliams and N. J. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 998. [8] H. Niederreiter, Point sets and seuence with small discrepancy, Monatsh. Math ) [9] H. Niederreiter, A combinatorial problem for vector spaces over nite elds, Discrete Math ) [0] H. Niederreiter, Orthogonal arrays and other combinatorial aspects in the theory of uniform point distributions in unit cubes, Discrete Math. 06/07 992) [] M. Yu. Rosenbloom and M. A. Tsfasman, Codes for m-metric, Problemy Peredachi Informatsii ), no, Russian). English translation in Problems Inform. transmission ), no, [2] M. M. Skriganov, Coding theory and uniform distributions, Algebra i Analiz 3 200), no 2, Russian). English translation to appear in St. Petesburg Math. J.